A110428 a(1) = 1 and a(2) = 2. Subsequent terms are generated like this: if a(m) is the last term available -- say a(2) -- then a(m+1) = a(m) * a(m-1), a(m+2) = a(m) * a(m-1) * a(m-2), ..., a(2*m-1) = a(m) * a(m-1) * a(m-2) * ... * a(2) * a(1), a(2*m) = a(2*m-1) * a(2*m-2), and so on.

1, 2, 2, 4, 4, 16, 32, 64, 64, 4096, 131072, 2097152, 8388608, 33554432, 67108864, 134217728, 134217728, 18014398509481984, 1208925819614629174706176, 40564819207303340847894502572032, 340282366920938463463374607431768211456

Offset

1,2

Comments

By choosing appropriate values for a(1) and a(2), many such sequences can be generated.

Links

Table of n, a(n) for n=1..21.

Formula

From Petros Hadjicostas, Nov 13 2019: (Start)

a(n) = a(n-1) * a(m) for n >= 3, where m = 2^(p+1) + 2 - n and p is the unique integer such that 2^p < n - 1 <= 2^(p+1), starting with a(1) = 1 and a(2) = 2.

a(A000051(n)) = a(2^n + 1) = a(2^n) for n >= 1.

a(A000051(n) + 1) = a(2^n + 2) = a(2^n + 1) * a(2^n) = a(2^n)^2 for n >= 1.

log[2](a(n)) = A329474(n) for n >= 1. (End)

Example

a(3) = a(2)*a(1) = 2. [Now a(3) is the last term available.]
a(4) = a(3)*a(2) = 4.
a(5) = a(3)*a(2)*a(1) = 4. [Now a(5) is the last term available.]
a(6) = a(5)*a(4) = 16.
a(7) = a(5)*a(4)*a(3) = 32.
a(8) = a(5)*a(4)*a(3)*a(2) = 64.
a(9) = a(5)*a(4)*a(3)*a(2)*a(1) = 64. [Now a(9) is the last term available.]
a(10) = a(9)*a(8) = 4096.
a(11) = a(9)*a(8)*a(7) = 131072.
...
a(17) = a(9)*a(8)*...*a(1) = 134217728. [Now a(17) is the last term available.]
a(18) = a(17)*a(16) = 18014398509481984.
[Example extended by Petros Hadjicostas, Nov 13 2019]

Maple

a := proc(n) option remember;
`if`(n < 3, [1, 2][n], a(n - 1) * a(2^ceil(log[2](n - 1)) + 2 - n));
end proc;
seq(a(n), n = 1..25); # Petros Hadjicostas, Nov 13 2019

Crossrefs

Cf. A000051 (index of "available" terms as described above), A050049 (an additive version of this sequence), A329474 (log[2] of this sequence).

Sequence in context: A112869 A086117 A095240 * A076928 A286737 A025557

Adjacent sequences: A110425 A110426 A110427 * A110429 A110430 A110431

Keyword

easy,nonn

Author

Amarnath Murthy, Aug 01 2005

Extensions

More terms from Joshua Zucker, May 10 2006

Status

approved